Theorem madebdayim 27884

Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

Ref	Expression
madebdayim	⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6868	. . 3 ⊢ (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ dom M )
2		madef 27832	. . . 4 ⊢ M :On⟶𝒫 No
3	2	fdmi 6673	. . 3 ⊢ dom M = On
4	1, 3	eleqtrdi 2846	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ On)
5		fveq2 6834	. . . . . 6 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
6		sseq2 3960	. . . . . 6 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
7	5, 6	raleqbidv 3316	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝑏)( bday ‘𝑥) ⊆ 𝑏))
8		fveq2 6834	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
9	8	sseq1d 3965	. . . . . 6 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
10	9	cbvralvw 3214	. . . . 5 ⊢ (∀𝑥 ∈ ( M ‘𝑏)( bday ‘𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏)
11	7, 10	bitrdi 287	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏))
12		fveq2 6834	. . . . 5 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
13		sseq2 3960	. . . . 5 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14	12, 13	raleqbidv 3316	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴))
15		elmade2 27854	. . . . . . . 8 ⊢ (𝑎 ∈ On → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
16	15	adantr 480	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17		elpwi 4561	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝒫 ( O ‘𝑎) → 𝑙 ⊆ ( O ‘𝑎))
18		elpwi 4561	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝒫 ( O ‘𝑎) → 𝑟 ⊆ ( O ‘𝑎))
19	17, 18	anim12i 613	. . . . . . . . . 10 ⊢ ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)))
20		unss 4142	. . . . . . . . . 10 ⊢ ((𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)) ↔ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎))
21	19, 20	sylib 218	. . . . . . . . 9 ⊢ ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎))
22		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23		simplll 774	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24		dfss3 3922	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ ( O ‘𝑎))
25		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → ( bday ‘𝑦) = ( bday ‘𝑧))
26	25	sseq1d 3965	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑧) ⊆ 𝑏))
27	26	rspccv 3573	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏))
28	27	ralimi 3073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → ∀𝑏 ∈ 𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏))
29		rexim 3077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
30	28, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
31	30	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
32		elold 27855	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏)))
33	32	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏)))
34		bdayon 27748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ‘𝑧) ∈ On
35		onelssex 6366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( bday ‘𝑧) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
36	34, 35	mpan 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
37	36	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
38	31, 33, 37	3imtr4d 294	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) → ( bday ‘𝑧) ∈ 𝑎))
39	38	ralimdv 3150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
40	24, 39	biimtrid 242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
41	40	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎)
42	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎)
43		bdayfun 27744	. . . . . . . . . . . . . . . . 17 ⊢ Fun bday
44		oldssno 27837	. . . . . . . . . . . . . . . . . . 19 ⊢ ( O ‘𝑎) ⊆ No
45		sstr 3942	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) ∧ ( O ‘𝑎) ⊆ No ) → (𝑙 ∪ 𝑟) ⊆ No )
46	44, 45	mpan2 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (𝑙 ∪ 𝑟) ⊆ No )
47		bdaydm 27746	. . . . . . . . . . . . . . . . . 18 ⊢ dom bday = No
48	46, 47	sseqtrrdi 3975	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (𝑙 ∪ 𝑟) ⊆ dom bday )
49		funimass4 6898	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun bday ∧ (𝑙 ∪ 𝑟) ⊆ dom bday ) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
50	43, 48, 49	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
51	50	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
52	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
53	42, 52	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎)
54		cutbdaybnd 27791	. . . . . . . . . . . . 13 ⊢ ((𝑙 <<s 𝑟 ∧ 𝑎 ∈ On ∧ ( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
55	22, 23, 53, 54	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56		fveq2 6834	. . . . . . . . . . . . 13 ⊢ ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘(𝑙 |s 𝑟)) = ( bday ‘𝑥))
57	56	sseq1d 3965	. . . . . . . . . . . 12 ⊢ ((𝑙 |s 𝑟) = 𝑥 → (( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑎))
58	55, 57	syl5ibcom 245	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘𝑥) ⊆ 𝑎))
59	58	expimpd 453	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎))
60	59	ex 412	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎)))
61	21, 60	syl5 34	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎)))
62	61	rexlimdvv 3192	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎))
63	16, 62	sylbid 240	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) → ( bday ‘𝑥) ⊆ 𝑎))
64	63	ralrimiv 3127	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎)
65	64	ex 412	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → ∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎))
66	11, 14, 65	tfis3 7800	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴)
67		fveq2 6834	. . . . 5 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
68	67	sseq1d 3965	. . . 4 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
69	68	rspccv 3573	. . 3 ⊢ (∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴 → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
70	66, 69	syl 17	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
71	4, 70	mpcom 38	1 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ∪ cun 3899   ⊆ wss 3901  𝒫 cpw 4554   class class class wbr 5098  dom cdm 5624   “ cima 5627  Oncon0 6317  Fun wfun 6486  ‘cfv 6492  (class class class)co 7358   No csur 27607   bday cbday 27609   <<s cslts 27753   |s ccuts 27755   M cmade 27818   O cold 27819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611  df-bday 27612  df-slts 27754  df-cuts 27756  df-made 27823  df-old 27824

This theorem is referenced by:  oldbdayim  27885  madebday  27896  onsbnd  28277  onsbnd2  28278